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A089087
Triangular array of coefficients multiplied by n! of polynomials in e. These give the expected number of trials needed for the sum of uniform random variables from the interval [0,1] to exceed n+1.
10
1, 1, -1, 2, -4, 1, 6, -18, 12, -1, 24, -96, 108, -32, 1, 120, -600, 960, -540, 80, -1, 720, -4320, 9000, -7680, 2430, -192, 1, 5040, -35280, 90720, -105000, 53760, -10206, 448, -1, 40320, -322560, 987840, -1451520, 1050000, -344064, 40824, -1024, 1, 362880, -3265920, 11612160, -20744640, 19595520
OFFSET
0,4
COMMENTS
Expected number of uniform random choices of X from interval[0,1] so that their sum exceeds ...
1 is e/0!,
2 is (e^2-e)/1!,
3 is (2e^3-4e^2+e)/2!.
REFERENCES
J. Derbyshire, "Prime Obsession: Bernhard Riemann and the Greatest Unsolved...", Henry Press, 2003, footnote on page 366.
J. V. Uspenski, "Introduction to Mathematical Probability", McGraw Hill, 1937, p. 278.
LINKS
Eric Weisstein's World of Mathematics, Uniform Sum Distribution.
FORMULA
T(n,k) = (-1)^k*n!*(n+1-k)^k/k!; k-th coefficient of n-th row for n >= 0 and k >= 0.
E.g.f.: 1/(exp(y*x)-x).
EXAMPLE
Triangle begins:
1,
1, -1,
2, -4, 1,
6, -18, 12, -1,
24, -96, 108, -32, 1,
120, -600, 960, -540, 80, -1,
720, -4320, 9000, -7680, 2430, -192, 1,
5040, -35280, 90720, -105000, 53760, -10206, 448, -1,
40320, -322560, 987840, -1451520, 1050000, -344064, 40824, -1024, 1,
362880, -3265920, 11612160, -20744640, 19595520, -9450000, 2064384, -157464, 2304, -1,
...
MATHEMATICA
f[n_] := Sum[(-1)^k*(n-k+1)^k*E^(n-k+1)/k!, {k, 0, n}]; (* f(0)=A001113=e, f(1)=A090142, f(2)=A090143, f(3)=A089139, f(4)=A090611 *)
Table[n!*CoefficientList[f[n], E] // Reverse // Most, {n, 0, 9}] // Flatten (* Jean-François Alcover, Nov 05 2013 *)
PROG
(SageMath)
def A089087_row(n):
R.<x> = ZZ[]
P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
return [(-1)^i*P[n-i+1] for i in (0..n)]
for n in (0..5): print(A089087_row(n)) # Peter Luschny, May 03 2013
KEYWORD
easy,sign,tabl
AUTHOR
Brian Dunfield (brian.dunfield(AT)sympatico.ca), Dec 04 2003
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Dec 05 2003
STATUS
approved